Nearest just interval: Difference between revisions

Revision as of 00:30, 9 November 2010

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The original revision id was 177738501.

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Original Wikitext content:

An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.

Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.

The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations|best relative approximation]]. Here it is required that |qr - p| is less than |nr - m| for any n < q. 

==Examples==
The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios:
|| **freq. ratio** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
|| ... || ... || ... || ... ||
||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0||
||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 ||
||= 4 / 3 ||= 3.585 ||= 498.04 ||= 101.96 ||
||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 ||
||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 ||
|| ... || ... || ... || ... ||

The 300-cent interval 2^(1/4) (3 steps of [[12edo]], "minor third") approximates following ratios:
|| **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
|| ... || ... || ... || ... ||
||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0||
||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 ||
||= 13 / 11 ||= 7.160 ||= 289.21 ||= 10.79 ||
||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 ||
||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 ||
|| ... || ... || ... || ... ||

Original HTML content:

<html><head><title>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.<br />
<br />
Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br />
<br />
The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that |qr - p| is less than |nr - m| for any n &lt; q. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2>
The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) approximates following ratios:<br />


<table class="wiki_table">
    <tr>
        <td><strong>freq. ratio</strong><br />
</td>
        <td><strong>log2(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
</td>
        <td><strong>size</strong> in cents<br />
</td>
        <td><strong>&quot;error&quot;</strong> in cents<br />
</td>
    </tr>
    <tr>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1 / 1<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">600.0<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3 / 2<br />
</td>
        <td style="text-align: center;">2.585<br />
</td>
        <td style="text-align: center;">701.96<br />
</td>
        <td style="text-align: center;">101.96<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4 / 3<br />
</td>
        <td style="text-align: center;">3.585<br />
</td>
        <td style="text-align: center;">498.04<br />
</td>
        <td style="text-align: center;">101.96<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7 / 5<br />
</td>
        <td style="text-align: center;">5.129<br />
</td>
        <td style="text-align: center;">582.51<br />
</td>
        <td style="text-align: center;">17.49<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17 / 12<br />
</td>
        <td style="text-align: center;">7.672<br />
</td>
        <td style="text-align: center;">603.00<br />
</td>
        <td style="text-align: center;">3.000<br />
</td>
    </tr>
    <tr>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
    </tr>
</table>

<br />
The 300-cent interval 2^(1/4) (3 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;minor third&quot;) approximates following ratios:<br />


<table class="wiki_table">
    <tr>
        <td><strong>freq. ratio</strong><br />
</td>
        <td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
</td>
        <td><strong>size</strong> in cents<br />
</td>
        <td><strong>&quot;error&quot;</strong> in cents<br />
</td>
    </tr>
    <tr>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1 / 1<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">0.0<br />
</td>
        <td style="text-align: center;">300.0<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6 / 5<br />
</td>
        <td style="text-align: center;">4.907<br />
</td>
        <td style="text-align: center;">315.64<br />
</td>
        <td style="text-align: center;">15.64<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13 / 11<br />
</td>
        <td style="text-align: center;">7.160<br />
</td>
        <td style="text-align: center;">289.21<br />
</td>
        <td style="text-align: center;">10.79<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19 / 16<br />
</td>
        <td style="text-align: center;">8.248<br />
</td>
        <td style="text-align: center;">297.51<br />
</td>
        <td style="text-align: center;">2.49<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25 / 21<br />
</td>
        <td style="text-align: center;">9.036<br />
</td>
        <td style="text-align: center;">301.84<br />
</td>
        <td style="text-align: center;">1.84<br />
</td>
    </tr>
    <tr>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
        <td>...<br />
</td>
    </tr>
</table>

</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-08 23:43:29 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-09 00:30:40 UTC</tt>.<br>
-: The original revision id was <tt>177732917</tt>.<br>
+: The original revision id was <tt>177738501</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive.
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.
 Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.
@@ Line 14: / Line 14: @@
 ==Examples==
 The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios:
-|| **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
+|| **freq. ratio** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
 || ... || ... || ... || ... ||
+||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0||
 ||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 ||
+||= 4 / 3 ||= 3.585 ||= 498.04 ||= 101.96 ||
 ||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 ||
 ||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 ||
@@ Line 24: / Line 26: @@
 || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
 || ... || ... || ... || ... ||
+||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0||
 ||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 ||
+||= 13 / 11 ||= 7.160 ||= 289.21 ||= 10.79 ||
 ||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 ||
 ||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 ||
@@ Line 30: / Line 34: @@
 </pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Nearest just interval&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An irrational interval or ratio of frequencies given by a real number r has an infinite list of &lt;em&gt;nearest just intervals&lt;/em&gt;; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call &lt;em&gt;best rational approximations&lt;/em&gt;. A ratio of integers p/q with q &amp;gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &amp;lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. &lt;br /&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Nearest just interval&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An irrational interval or ratio of frequencies given by a real number r has an infinite list of &lt;em&gt;nearest just intervals&lt;/em&gt;; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call &lt;em&gt;best rational approximations&lt;/em&gt;. A ratio of integers p/q with q &amp;gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &amp;lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.&lt;br /&gt;
 &lt;br /&gt;
 Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.&lt;br /&gt;
@@ Line 44: / Line 48: @@
          &lt;td&gt;&lt;strong&gt;freq. ratio&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;log(&lt;a class="wiki_link" href="/Tenney%20Height"&gt;Tenney Height&lt;/a&gt;)&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;log2(&lt;a class="wiki_link" href="/Tenney%20Height"&gt;Tenney Height&lt;/a&gt;)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;size&lt;/strong&gt; in cents&lt;br /&gt;
@@ Line 59: / Line 63: @@
 &lt;/td&gt;
          &lt;td&gt;...&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1 / 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;600.0&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 67: / Line 81: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;701.96&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;101.96&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;4 / 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;3.585&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;498.04&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;101.96&lt;br /&gt;
@@ Line 126: / Line 150: @@
 &lt;/td&gt;
          &lt;td&gt;...&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1 / 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0.0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;300.0&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 136: / Line 170: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;15.64&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;13 / 11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7.160&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;289.21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;10.79&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

Nearest just interval: Difference between revisions

Revision as of 00:30, 9 November 2010

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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